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1

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

Statistics 2

The central limit theorem and sampling distributions

Abraham de Moivre French Hugenot refugee in London originator of the Central Limit Theorem

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2

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research Central limit theorem

Statistics Introduction 2

The Central Limit TheoremOur evaluation of a t score for statistical significance depends on sample size

Larger samples yield more ldquonormalrdquo tighter distributions (less error variancehellip)

With smaller samples we use more conservative assumptions about the sampling distribution

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3

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Psychology 242 Dr McKirnan

-3 -2 -1 0 +1 +2 +3

t Scores

3413 of

cases

3413 of

cases

1359 of

cases

225 of

cases

1359 of

cases

225 of

cases

The normal distribution

Here is the Sampling DistributionThis is the normal distribution segmented into t units (similar to Z units or Standard Deviations)

Each t unit (eg between t = 0 and t = 1) represents a fixed percentage of cases

Central Limit Theorem our assumptions about t values have to change depending upon the size of our sample

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4

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

The Central Limit Theorem small samples

lt-- smaller M larger ---gt

True Population M ldquoTruerdquo normal

distribution

ScoreScore Score

ScoreScore

Score Score Score

Score

Score

Score

With few scores in the sample a few extreme or ldquodeviantrdquo values have a large effect

The distribution is ldquoflatrdquo or has high variance

Central Limit Theorem

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5

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

The Central Limit Theorem larger samples

With more scores the effect of extreme or ldquodeviantrdquo values is offset by other values

Central Limit Theorem

The distribution has less variance amp is more normal

ScoreScoreScore Score

ScoreScore

Score Score

Score

Score

lt-- smaller M larger ---gt

True Population M ldquoTruerdquo normal

distribution

Score

Score ScoreScore

Score Score ScoreScoreScoreScore

Score

Score

ScoreScore

ScoreScore

Score

Score

Score Score

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6

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

The Central Limit Theorem large samples

Central Limit Theorem

ScoreScoreScore Score

ScoreScore

Score Score

Score

Score

lt-- smaller M larger ---gt

True Population M ldquoTruerdquo normal

distribution

Score

Score ScoreScore

Score ScoreScoreScoreScore

Score

Score

Score

ScoreScore

ScoreScore

Score

Score

Score Score

ScoreScore

Score

Score Score

Score

Score

ScoreScore

ScoreScore

Score

Score

Score

Score

Score

Score

Score

ScoreScore

Score

ScoreScore With many scores ldquodeviantrdquo values are completely offset by other values

The distribution is normal with low(er) variance

The sampling distribution better approximates the population distribution

Pascalrsquos quincunx demonstration is at httpwwwmathsisfuncomdataquincunxhtml

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7

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

Central limit theorem amp evaluating t scores

1 If the groups are small the M score for each group reflects a lot of error variance

2 This increases the likelihood that error variance not an experimental effect led to differences between Ms

3 Since smaller samples (lower df) = more variance t must be larger for us to consider it statistically significant (lt 5 likely to have occurred by chance alone)

4 We evaluate t vis-agrave-vis a sampling distribution based on the df for the experiment

5 Critical value for t with p lt05 thus goes up or down depending upon sample size (df)

The same logic applies with samples we use to test hypotheses

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8

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2lt-- smaller M larger ---gt

The Central Limit Theorem small samples

Central Limit Theorem applied to a sampling distribution How well do small samples reflect the ldquotruerdquo population

Since small samples have a lot of error a distribution of small samples is relatively

ldquoflatrdquo (lot of variance)hellip

M(n=10)

M(n=10) M(n=10)M(n=10)

M(n=10)M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)M(n=10)

M(n=10)

M(n=10)

M(n=10)M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)

M(n=10)M(n=10)

M(n=10)

M(n=10)M(n=10)M(n=10) M(n=10)

M(n=10)M(n=10)

M(n=10) M(n=10)

M(n=10)

M(n=10)

M of sample Ms (approximates population M)

M(n=10)

M(n=10) M(n=10)

Imagine we calculate the M for each of 50 samples each n=10

Many sample Ms may be far from the M of

sample Ms

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9

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

The Central Limit Theorem larger samples

Central Limit Theorem amp sampling distributions larger samples

The M for each sample has less error (since it

has larger n) so the distribution will be ldquocleanerrdquo and more

normal

M(n=25)

M(n=25) M(n=25)

M(n=25)

M(n=25)M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)M(n=25)

M(n=25)

M(n=25)

M(n=25)M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)

M(n=25)M(n=25)

M(n=25)M(n=25)M(n=25) M(n=25)

M(n=25)M(n=25)

M(n=25) M(n=25)

M(n=25)

M(n=25)

lt-- smaller M larger ---gt

lsquoTruerdquo M of sample Ms

M(n=25)

M(n=25) M(n=25)

Now we collect another 50 samples but each n=25

It is less likely that any individual sample M would be far from the M of sample Ms

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10

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

The Central Limit Theorem larger samples

Central Limit Theorem amp sampling distributions large samples

Since each individual sample has low error a

distribution of large sample Ms will have

low variance

M(n=50)

M(n=50) M(n=50)

M(n=50)

M(n=50)M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)M(n=50)

M(n=50)

M(n=50)

M(n=50)M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)

M(n=50)M(n=50)

M(n=50)M(n=50)M(n=50)

M(n=50)

M(n=50)M(n=50)

M(n=50) M(n=50)

M(n=50)

M(n=50)

lt-- smaller M larger ---gt

lsquoTruerdquo M of sample Ms

M(n=50)

M(n=50)M(n=50)

Our third set of samples are each fairly large say n=50

It is unlikely for a sample M to far exceed the M of the sample Ms by chance alone

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11

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

Central limit theorem critical values

Central limit theorem When df gt 120 we assume a perfectly normal distribution

(Here Z = t no compensation for sample size)

With smaller samples we assume more error in each group

When df lt 120 we use t to estimate a sampling distribution based on the total df (ie ns of groups being sampled)

Alpha [ α ] Probability criterion for ldquostatistical significancerdquo typically p lt 05

Critical value Cut off point for alpha on distribution

With df gt 120 critical value for plt05 = + 198 (Z = t)

With df lt 120 we adjust the critical value based on the sampling distribution we use

As df goes down we assume a more conservative sampling distribution and use a larger critical value for p lt05

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12

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

Sampling Distributions and Critical Values

-2 -1 0 +1 +2 Z Score

(standard deviation units) 24 of cases gt +19824 of cases lt -198

Critical value for plt05 = 198 95 of cases (critical ratios differences between Ms) are lt +198 and gt -198

This sampling distribution n gt 120

Other graphs will show what happens as sample size decreasesZ or t (120) gt + 198

will occur by chance lt 5 of the time

A distribution with n gt 120 is ldquonormalrdquo

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13

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research Sampling distributions Critical Values when df = 18

-2 -1 0 +1 +2 Z Score

(standard deviation units)

With a smaller df we estimate a flatter more ldquoerrorfulrdquo curve

At df = 18 the critical value for plt05 = 210 a more conservative test

24 of cases gt +21024 of cases lt -210

Here group sizes are small

Group1 n = 10Group2 n = 10df = (10-1) + (10-1) = 18

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14

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

Critical Values n = 10

-2 -1 0 +1 +2 Z Score

(standard deviation units)

With only 8 df we estimate a flat conservative curve

Here the critical value for plt05 = 230

24 of cases gt +23024 of cases lt -230

This sampling distribution assumes 10 participants Group1 n = 5 Group2 n = 5 df = (5-1) + (5-1) = 8

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15

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Statistics Introduction 2

Central Limit Theorem variations in sampling distributions

24 of cases below this value 24 of cases above this value

-2 -1 0 +1 +2 Z Score

(standard deviation units)

N gt 120 t gt + 198 plt05

df = 18 t gt + 210 plt05

df = 8 t gt + 230 plt05

As samples sizes (df) go down the estimated sampling distributions of t scores based on them have more variance giving a more ldquoflatrdquo distribution

This increases the critical value for plt05

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16

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

1860 2306 2896 3355 5041

1833 2262 2821 3250 4781

1812 2228 2764 3169 4587

1796 2201 2718 3106 4437

1782 2179 2681 3055 4318

1771 2160 2650 3012 4221

1761 2145 2624 2977 4140

1753 2131 2602 2947 4073

1725 2086 2528 2845 3850

1708 2060 2485 2787 3725

1697 2042 2457 2750 3646

1684 2021 2423 2704 3551

1671 2000 2390 2660 3460

1658 1980 2358 2617 3373

1645 1960 2326 2576 3291

A t-table contains

df8 9

1011121314152025304060

120

Degrees of freedom (df) Size of the research samples (ngroup1 - 1) + (ngrp2 - 1)

Alpha levels

likelihood of a t occurring by chance

Alpha Levels

010 005 002 0010001

Critical Values Value t must exceed to be statistically significant [not occurring by chance] at a given alpha

Critical values of t

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17

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

1860 2306 2896 3355 5041

1833 2262 2821 3250 4781

1812 2228 2764 3169 4587

1796 2201 2718 3106 4437

1782 2179 2681 3055 4318

1771 2160 2650 3012 4221

1761 2145 2624 2977 4140

1753 2131 2602 2947 4073

1725 2086 2528 2845 3850

1708 2060 2485 2787 3725

1697 2042 2457 2750 3646

1684 2021 2423 2704 3551

1671 2000 2390 2660 3460

1658 1980 2358 2617 3373

1645 1960 2326 2576 3291

Critical values of t (2 tailed test)

Critical value of t is read across the row for the df in your study to the column for your alpha

p lt 05 is the most typical alpha

lower alpha (02 001 a more conservative test) requires a higher critical value

Critical values of t

Alpha = 05 df = 10

Alpha = 02 df = 13

Alpha = 05 df = 120

df

8 9

1011121314152025304060

120

Alpha Levels

010 005 002 0010001

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18

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research

Determining If A Result Is Statistically Significant

Assumptions

Null hypothesis the difference between Ms [or the correlation chi square etc] is gt 0 or lt 0 by chance alone

Statistical question is the effect in your experiment different from 0 by more than chance alone

More than chance alone is lt 5 of the time [p lt 05]

Steps

1 Derive the t value for the difference between groups

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19

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research Statistical significancehellip

2 Figure out what distribution to compare your t value to

bull Use the degrees of freedom (df) for this

bull df = (ngroup1 - 1) + (ngroup2 - 1)

bull The Central Limit Theorem tells us to assume there is more error (a more flat distribution) as df go down

4 Use the usual criteria [alpha value] for ldquostatistical significancerdquo of p lt 05 (unless you have good reason to use anotherhellip)

5 Find the value on the t table that corresponds to your df at your alpha This is the critical value that your t must exceed to be considered ldquostatistically significantrdquo

6 Compare your t to the critical value using the absolute value of t

Steps cont

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20

Dr David McKirnan davidmckuicedu

Psychology 242Introductionto Research Testing t

Statistics Introduction 2

1860 2306 2896 3355 5041

1833 2262 2821 3250 4781

1812 2228 2764 3169 4587

1796 2201 2718 3106 4437

1782 2179 2681 3055 4318

1771 2160 2650 3012 4221

1761 2145 2624 2977 4140

1753 2131 2602 2947 40731734 2101 2552 2878 3922

1725 2086 2528 2845 3850

1708 2060 2485 2787 3725

1697 2042 2457 2750 3646

1684 2021 2423 2704 3551

1671 2000 2390 2660 3460

1658 1980 2358 2617 3373

1645 1960 2326 2576 3291

df8 9

101112131415182025304060

120

Alpha Levels010 005 002 001

0001

bull Use p lt 05 (unless you want to be more conservative by using a higher value)

bull Look up your df to see what sampling distribution to compare your results to

bull With n = 10 per group df = (10-1) + (10-1) = 18

bull Compare your t to the critical value from the table

bull If the absolute value of t gt the critical value your effect is statistically significant at p lt 05